Rutgers Discrete Mathematics Seminar

Title: Q-polynomial distance-regular graphs and the DAHA of rank one

Speaker: Jae-Ho Lee, U. Wisconsin

Date: Tuesday, November 27, 2012 2:00pm

Location: Hill Center, Room 525, Rutgers University, Busch Campus, Piscataway, NJ


Let G denote a Q-polynomial distance-regular graph with vertex set X. We assume that G has q-Racah type and contains a Delsarte clique C. Fix a vertex x in C. We partition X according to the path-length distance to both x and C. This is an equitable partition. For each cell in this partition, consider the corresponding characteristic vector. These characteristic vectors form a basis for a vector space W over complex field. The universal DAHA of type (C^{vee}_1, C_1) is the algebra H_q defined by generators {t_n^{pm1}}^3_{n=0} and relations (i) t_nt^{-1}_n = t^{-1}_nt_n = 1; (ii) t_n+t_n^{-1} is central; (iii) t_0t_1t_2t_3 = q^{-1/2}. We display an H_q-module structure for W. For this module and up to affine transformation, (i) t_0t_1+(t_0t_1)^{-1} acts as the adjacency matrix of G; (ii) t_3t_0+(t_3t_0)^{-1} acts as the dual adjacency matrix of G with respect to C; (iii) t_1t_2+(t_1t_2)^{-1} acts as the dual adjacency matrix of G with respect to x.